Free algebra over the intersection of varieties

Question

I am wondering if given to varieties $V,W$ in the sense of Universal algebra once could describe free algebras over $V\cap W$ in some meaningful way? Now what I mean by meaningful is not even completely clear to me. I would like to use this description to say something regarding the subalgebras of $F_{V\cap W}(x_1,...,x_n)$ if I know something about subalgebras of $F_{V}(x_1,...,x_n)$ and $F_{W}(x_1,...,x_n)$. I am aware this isn't completely well posed question but maybe someone is able to help.